Definition
The owner, U20, quoting the exact intertwining ΦA_i = B_iΦ: “I think presenting this as an equivalence might be too constraining. I agree that there should be some transport map that at least gets to within some eps, but that’s just a similarity constraint, not an equivalence. This is why the game ‘telephone’ works.” A50: local discrepancy δ_i(s) = d_t(Φ(A_i(s)), B_i(Φ(s))) with E_μ[δ_i] ≤ ε_i, per operation; directed distortion D(u⇝v) ≠ D(v⇝u); fidelity of a stochastic translation process F(T; u, c) = E_(v∼T)[Sim_O(u, v)]. “define fidelity as a graded quantity, not an equivalence relation.”
What it is not
Not an equivalence, approximate or otherwise (TERM-033): “Even approximate equivalence may be the wrong organizing concept if the phenomenon we’re interested in is graded preservation under successive transformations.” Equivalence is recovered “as a special case—for example, zero distortion under a suitably separating observation family”. Manuscript §12: “stochastic transport generalizes exact equivalence to graded fidelity.”